scholarly journals On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders

Entropy ◽  
2007 ◽  
Vol 9 (3) ◽  
pp. 118-131 ◽  
Author(s):  
Haidong Liu ◽  
Prabhamani Patil ◽  
Uichiro Narusawa
Keyword(s):  
Viscous Flow ◽  
Brinkman Equation ◽  
Parallel Plates

Viscous flow in a channel with periodic cross-bridging fibres: exact solutions and Brinkman approximation

1991 ◽  
Vol 226 ◽  
pp. 125-148 ◽  
Author(s):  
Ruey-Yug Tsay ◽  
Sheldon Weinbaum
Keyword(s):  
Aspect Ratio ◽  
Drag Coefficient ◽  
Viscous Flow ◽  
Velocity Component ◽  
Stokes Equations ◽  
Interpolation Formula ◽  
Numerical Results ◽  
Brinkman Equation ◽  
Capillary Endothelium ◽  
Channel Height

A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compared with the solution of the Brinkman equation for the flow through a porous medium in a channel. This comparison shows that the Brinkman approximation is very good when B > 5, but breaks down when B [les ] O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficient f, and the Darcy permeability Kp are presented. It is shown that the velocity component perpendicular to the parallel walls is only significant within the viscous layers surrounding the fibres, whose thickness is of the order of half the channel height B′. One finds that when the aspect ratio B > 5, the neglect of the vertical velocity component vz can lead to large errors in the satisfaction of the no-slip boundary conditions on the surfaces of the fibres and large deviations from the approximate solution in Lee (1969), in which vz and the normal pressure field are neglected. The numerical results show that the drag coefficient of the fibrous bed increases dramatically when the open gap between adjacent fibres Δ′ becomes smaller than B′. The predictions of the new theory are used to examine the possibility that a cross-bridging slender fibre matrix can exist in the intercellular cleft of capillary endothelium as proposed by Curry & Michel (1980).

Viscous flow around three-dimensional macroscopic cavities in a granular material

2021 ◽  
Vol 931 ◽  
Author(s):  
Osamu Sano ◽  
Timir Karmakar ◽  
G.P. Raja Sekhar
Keyword(s):  
Granular Material ◽  
Viscous Flow ◽  
Stokes Equation ◽  
Three Dimensional ◽  
Volume Flow ◽  
The Other ◽  
Brinkman Equation ◽  
Local Enhancement ◽  
Reasonable Estimate ◽  
The Stokes Equation

Viscous flow around spherical macroscopic cavities in a granular material is investigated. The Stokes equation inside and the Darcy–Brinkman equation outside the cavities are considered. In particular, the interaction of two equally sized cavities positioned in tandem is examined in detail, where the asymptotic effect of the other cavity is taken into account. The present analysis gives a reasonable estimate on the volume flow into the cavity and the local enhancement of stresses. This is applicable to predict the microscale waterway formation in that material, onset of landslides, collapse of cliffs and river banks, etc.

Axisymmetric motion of a spherical porous particle perpendicular to two parallel plates with slip surfaces

2015 ◽  
Vol 93 (7) ◽  
pp. 784-795 ◽  
Author(s):  
E.I. Saad
Keyword(s):  
Fluid Motion ◽  
Brinkman Equation ◽  
Parallel Plates ◽  
Porous Sphere ◽  
Field Equations ◽  
Clear Fluid ◽  
Stress Jump ◽  
Spherical Coordinate ◽  
Good Convergence ◽  
Special Cases

A combined analytical–numerical approach to the problem of the low Reynolds number motion of a porous sphere normal to one of two infinite parallel plates at an arbitrary position between them in a viscous fluid is investigated. The clear fluid motion governed by the Stokes equation and the Darcy–Brinkman equation is used to model the flow inside the porous material. The motion in each of the homogeneous regions is coupled with the continuity of the velocity components, the continuity of the normal stress, and the tangential stress jump condition. The fluid is allowed to slip at the surface of the walls. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The collocation solutions for the hydrodynamic interactions between the porous sphere and the walls are calculated with good convergence for various values of the slip coefficient of the walls, the separation between the porous sphere and the walls, the stress jump coefficient, and a coefficient that is proportional to the permeability. For the special cases of a solid sphere, our drag results show excellent agreement with the available solutions in the literature for all relative particle-to-wall spacing.

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